The Time Variation in Risk Appetite and Uncertainty

62 Pages·2017·1.05 MB·English

The Time Variation in Risk Appetite and Uncertainty
Geert Bekaert Eric C. Engstrom Nancy R. Xu∗
November 10, 2017
*PRELIMINARY*
Abstract
We develop new measures of time-varying risk aversion and economic uncertainty that
can be calculated from observable financial information at high frequencies. Our approach
hasfourimportantelements. First,weformulateadynamicno-arbitrageassetpricingmodel
that consistently prices all assets under assumptions regarding the joint dynamics among
asset-specific cash flow dynamics, macroeconomic fundamentals and risk aversion. Second,
both the fundamentals and cash flow dynamics feature time-varying heteroskedasticity and
non-Gaussianitytoaccommodatedynamicsobservedinthedata, whichwedocument. This
allows us to distinguish time variation in economic uncertainty (the amount of risk) from
time variation in risk aversion (the price of risk). Third, despite featuring non-Gaussian
dynamics, the model retains closed-form solutions for asset prices. Fourth, our approach
exploits information on realized volatility and option prices for the two main risky asset
classes,equitiesandcorporatebonds,tohelpidentifyanddifferentiateeconomicuncertainty
from risk aversion. We find that equity variance risk premiums are very informative about
risk aversion, whereas credit spreads and corporate bond volatility are highly correlated
with economic uncertainty. Model-implied risk premiums beat standard instrument sets
predicting excess returns on equity and corporate bonds. A financial proxy to our economic
uncertainty predicts output growth negatively and significantly, even in the presence of the
VIX.
∗BekaertiswithColumbiaUniversityandtheNBER,EngstromiswiththeFederalReserveBoardofGover-
nors, Xu is with Columbia University. The views expressed in this document do not necessarily reflect those of
theFederalReserveSystem,itsBoardofGovernors,orstaff. Allerrorsarethesoleresponsibilityoftheauthors.
Contact: Xu ([email protected])
1 Introduction
It has become increasingly commonplace to assume that changes in risk appetites are an
important determinant of asset price dynamics. For instance, the behavioral finance literature
(see, e.g., Lemmon and Portnaiguina (2006) and Baker and Wurgler (2006) for a discussion)
has developed “sentiment indices,” and there are now a wide variety of “risk aversion” or
“sentiment” indicators available, created by financial institutions (see Coudert and Gex (2008)
for a survey). The “structural” dynamic asset pricing literature has meanwhile proposed time-
varying risk aversion as a potential explanation for salient asset price features (see Campbell
andCochrane(1999)andalargenumberofrelatedarticles), whereasreduced-formassetpricing
models, aiming to simultaneously explaining stock return dynamics and option prices, have also
concludedthattime-varyingpricesofriskareimportantdriversofstockreturnandoptionprice
dynamics (see Bakshi and Wu, 2010; Bollerslev, Gibson, and Zhou, 2011; Broadie, Chernov,
andJohannes, 2007). Riskaversionhasalsofeaturedprominentlyinrecentmonetaryeconomics
papers that suggest a potential link between loose monetary policy and the risk appetite of
market participants, spurring a literature on what structural economic factors would drive risk
aversionchanges(see,e.g.,Rajan,2006; AdrianandShin,2009; Bekaert,Hoerova,andLoDuca,
2013). Ininternationalfinance,Miranda-AgrippinoandRey(2015)andRey(2015)suggestthat
global risk aversion is a key transmission mechanism for US monetary policy to be exported to
countries worldwide and is a major source of asset return comovements across countries (see
also Xu, 2017). Finally, several papers on sovereign bonds (e.g. Bernoth and Erdogan, 2012)
have stressed the importance of global risk aversion in explaining their dynamics and contagion
across countries.
Our goal is to develop a measure of time-varying risk aversion that is relatively easy
to estimate and compute, so that it can be compared to other indices and tracked over time.
However,themeasureshouldalsocorrectfordeficienciesplaguingmanyofthecurrentmeasures.
First, it must control for macro-economic uncertainty; we want to separately identify both the
aversion to risk (the price of risk) and the amount of risk. To do so, we build on dynamic
asset pricing theory. Essentially, our risk aversion measure constitutes a second factor in the
pricing kernel that is not driven by macroeconomic fundamentals. The modeling framework
therefore is related, but not identical, to the habit models of Campbell and Cochrane (1999),
Menzly, Santos and Veronesi (2004) and Wachter (2006). As in Bekaert, Engstrom and Xing
(2009) and Bekaert, Engstrom and Grenadier (2010), we allow for a stochastic risk aversion
component that is not perfectly correlated with fundamentals. As an important byproduct,
we also derive a measure of economic uncertainty, which constitutes an alternative to recent
measures (e.g. Juardo, Ludvigson, and Ng, 2015). In the model, asset prices are linked to cash
flow dynamics and preferences in an internally consistent fashion. In contrast, a number of
articles develop time-varying risk aversion measures motivated by models that really assume
“constant” prices of risk and hence are inherently inconsistent (see, for example, Bollerslev,
Gibson, and Zhou, 2011), or fail to fully model the link between fundamentals and asset prices
(see e.g. Bekaert and Hoerova, 2016). Third, as is well-known, asset prices and returns display
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dynamics with highly non-Gaussian distributions that are time varying. In fact, a number of
articles (see Bollerslev and Todorov, 2011; Liu, Pan, and Wang, 2004; Santa-Clara and Yan,
2010) suggest that compensation for rare events (“jumps”) accounts for a large fraction of
equity risk premiums. To accommodate these non-linearities in a tractable fashion, we use
the Bad Environment-Good Environment (BEGE, henceforth) framework developed in Bekaert
and Engstrom (2017). Shocks are modeled as the sum of two variables with de-meaned gamma
distributions, whose shape parameters vary through time. The model delivers conditional non-
Gaussian shocks, with changes in “good” or “bad” volatility also changing the conditional
distributionoftheprocess. Finally, ourdataincludemacroeconomicfundamentals, assetprices,
and options prices. The dynamic asset pricing and options literatures indirectly reveal the
difficulty in interpreting many existing risk aversion indicators. Often they use information
such as the VIX or return risk premiums that are obviously driven by both the amount of risk
and risk aversion. Disentangling the two is not straightforward. Articles such as Drechsler and
Yaron (2008), Bollerslev et al. (2009) and Bekaert and Hoerova (2016) point towards the use
of the VIX in combination with the (conditional) expected variance as particularly informative
about risk preferences. Therefore, this paper is also related to the literature on extracting
information about risk and risk preferences from option prices (for a survey, see Gai and Vause,
2006).
The use of different asset classes in deriving a single measure of risk aversion imposes the
important assumption that different markets are priced in an integrated setting. This may not
(always) bethe case. Duringthe 2007-2009global crisis, it was widelyrecognized thatarbitrage
opportunities surfaced between asset classes and sometimes within an asset class (for instance,
between Treasury bonds of different maturities, see e.g. Hu, Pan, and Wang, 2013). There may
well be a link between risk aversion and the existence of arbitrage opportunities. That is, in
uncertain, risk averse times, there is insufficient risky capital available, which causes different
asset classes to be priced incorrectly (see, for example, Gilchrist, Yankov, and Zakrajsek, 2009).
While consistent pricing across risky asset classes is a maintained assumption in our benchmark
model, we can easily test for consistent pricing by examining risk aversion measures implied by
different asset classes. We provide an example by comparing risk aversions filtered from risky
assets only and from both risky assets and Treasury bonds.
The remainder of the paper is organized as follows. Sections 2 and 3 presents the model
and estimation strategy in detail. Section 4 briefly outlines the data we use. Section 5 extracts
risk aversion and uncertainty from asset prices and discusses the links between the risk aversion
estimates and various financial variables. We also examine the behavior of the indices around
the Bear Stearns and Lehman Brothers bankruptcies. In Section 6, we link our measures of
risk appetite and uncertainty to alternative indices including ones produced by practitioners.
In Section 7, we discuss the case of risk aversion involving Treasury bonds. Concluding remarks
are in Section 8.
2
2 Modeling Risk Appetite and Uncertainty
In this section, we first define our concept of risk aversion in general terms in Section 2.1.
We then build a dynamic model with stochastic risk aversion and macro-economic factors af-
fecting the cash flows processes of two main risky asset classes, corporate bonds and equity.
The state variables are described in Section 2.2 and the pricing kernel in Section 2.3.
2.1 General Strategy
An ideal measure of risk aversion would be model free and not confound time variation
in economic uncertainty with time variation in risk aversion. There are many attempts in the
literaturetoapproximatethisideal, butinvariablyvariousmodelingandstatisticalassumptions
are necessary to tie down risk aversion. For example, in the options literature, a number
of articles (Aıt-Sahalia and Lo, 2000; Engle and Rosenberg, 2002; Jackwerth, 2000; Bakshi,
Kapadia and Madan, 2003; Britten-Jones and Neuberger, 2000) appear at first glance to infer
risk aversion from equity options prices in a general fashion, but it is generally the case that the
utility function is assumed to be of a particular form and/or to depend only on stock prices.
Another strand of the literature relies on general properties of pricing kernels. Using a
strictlypositivepricingkernelorstochasticdiscountfactor,M ,no-arbitrageconditionsimply
t+1
that for all gross returns, R,
E [M R ] = 1 (1)
t t+1 t+1
It is then straightforward to derive that any asset’s expected excess return can be written as
an asset specific risk exposure (“beta”, or β ) times a price of risk (or λ ), which applies to all
t t
assets (see also Coudert and Gex, 2008):
E [R ]−Rf = β λ (2)
t t+1 t t t
where Rf is the risk free rate, β = −Covt(Rt+1,Mt+1), and λ = Vart(Mt+1).
t t Vart(Mt+1) t Et(Mt+1)
Unfortunately, this price of risk is not equal to time-varying risk aversion, and in particu-
lar may confound economic uncertainty with risk aversion. In a simple power utility framework,
it is easy to show that the price of risk is linked to both the coefficient of relative risk aversion
and the volatility of consumption growth, the latter being a reasonable measure of economic
uncertainty.
Our approach is to start from a fairly general utility function defined over both funda-
mentals and non-fundamentals. Our measure of risk aversion simply is then the coefficient of
relative risk aversion implied by the utility function. We specify a fairly general consumption
process accommodating time variation in economic uncertainty and use the utility framework
to price assets, given general processes for the cash flows of assets. Therefore, while certainly
not model free, our risk aversion process is consistent with a wide set of economic models that
respect no-arbitrage conditions. Moreover, we can use any risky asset for which we can model
cash flows to help identify risk aversion. The identification of the risk aversion process takes
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into account that economic uncertainty varies through time and controls for non-Gaussianities
in cash flow processes.
Consider a period utility function in the HARA class:
(cid:16) (cid:17)1−γ
(cid:18) (cid:19) C
C Q
U = (3)
Q 1−γ
where C is consumption and Q is a process that will be shown to drive time-variation in risk
aversion. Essentially, when Q is high, consumption delivers less utility and marginal utility
increases. For the general HARA class of utility functions,
(cid:18)a b (cid:19)−1
Q = − = f(C) (4)
γ C
where a and γ are positive parameters, and b is an exogenous benchmark parameter or process.
Note that γ (the curvature parameter) is not equal to risk aversion in this framework. In
principle, all parameters (a, γ, b) could have time subscripts, but we only allow time-variation
in b. Note that the Q process depends on consumption, but we do not allow b to depend on
consumption. This excludes internal habit models, for example.
The coefficient of relative risk aversion for this class of models is given by
CU(cid:48)(cid:48)(C)
RRA = − = aQ (5)
U(cid:48)(C)
(cid:16) (cid:17)−2
and is thus proportional to Q. Note that dQ = −b aC −b < 0; in good times when
dC γ
consumption increases, risk aversion decreases.
For pricing assets, we need to derive the log pricing kernel which is the intertemporal
marginal rate of substitution in a dynamic economy. We assume an infinitely lived agent, facing
a constant discount factor of β, and the HARA period utility function given above. The pricing
kernel is then given by
(cid:20)U(cid:48)(C )(cid:21)
t+1
m = ln(β)+ln = ln(β)−γ∆c +γ∆q (6)
t+1 U(cid:48)(C ) t+1 t+1
t
where we use t to indicate time, lower case letters to indicate logs of uppercase variables, and
∆ to indicate log changes.
To get more intuition for this framework, note that the Campbell and Cochrane (1999)
(CC henceforth) utility function is a special case. CC use an external habit model, with utility
being a power function over C −H , where H is the habit stock. Of course, we can also write
t t t
C
t
C −H = (7)
t t
Q
t
with Q = Ct . So the CC utility function is a special case of our framework with a = γ and
t Ct−Ht
b = H. As C gets closer to the habit stock, risk aversion increases. Q is thus the inverse of
t t
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the surplus ratio in the CC article. CC also model q exogenously but restrict the correlation
t
between q and ∆c to be perfect. The “moody investor” economy in Bekaert, Engstrom, and
t t
Grenadier (2010) is also a special case. In that model, q is also exogenously modeled, but
t
has its own shock; that is, there are preference shocks not correlated with fundamentals. In
our general quest to identify risk aversion, we surely must allow for such shocks to hit q as
well. The model in Brandt and Wang (2003) is also a special case but the risk aversion process
specifically depends on inflation in addition to consumption growth. In fact, DSGE models in
macro-economics routinely feature preference shocks (see e.g. Besley and Coate, 2003).
In sum, our approach specifies a stochastic process for q (risk aversion), which constitutes
a second factor in the pricing kernel that is not fully driven by fundamentals (consumption
growth).
2.2 Economic Environment: State Variables
2.2.1 Macroeconomic Factors
Incanonicalassetpricingmodelsagentshaveutilityoverconsumption,butitiswellknown
thatconsumptiongrowthandassetreturnsshowverylittlecorrelation. Moreover, consumption
data are only available at the quarterly frequency. Because the use of options data is key to
our identification strategy and these data are only available since 1986, it is important to use
macro-economic data that are available at the monthly frequency. We therefore chose to use
industrial production, which is available at the monthly frequency, as our main macroeconomic
factor. In the macro-economic literature, much attention has been devoted recently to the
measurement of “real” uncertainty (see e.g. Jurado, Ludvigson and Ng, 2015) and its effects on
the real economy (see e.g. Bloom, 2009). We add to this literature by using a novel econometric
framework to extract two macro risk factors from industrial production: “good” uncertainty,
denoted by p , and “bad” uncertainty, denoted by n .
t t
Specifically,thechangeinlogindustrialproductionindex,θ ,hastime-varyingconditional
t
moments governed by two state variables: p and n . The conditional mean is modeled as a
t t
persistent process to accommodate a time-varying long-run mean of output growth:
θ = θ+ρ (θ −θ)+m (p −p)+m (n −n)+uθ , (8)
t+1 θ t p t n t t+1
where the growth shock is decomposed into two independent centered gamma shocks,
uθ = σ ω −σ ω . (9)
t+1 θp p,t+1 θn n,t+1
The shocks follow centered gamma distributions with time-varying shape parameters,
ωp,t+1 ∼ Γ(cid:101)(pt,1) (10)
ωn,t+1 ∼ Γ(cid:101)(nt,1), (11)
where Γ(cid:101)(x,1) denotes a centered gamma distribution with shape parameter x and a unit scale
5
parameter. The shape factors, p and n , follow autoregressive processes,
t t
p = p+ρ (p −p)+σ ω (12)
t+1 p t pp p,t+1
n = n+ρ (n −n)+σ ω , (13)
t+1 n t nn n,t+1
whereρ denotestheautoregressivetermofprocessx ,σ thesensitivitytoshockω ,and
x t+1 xx x,t+1
(cid:104) (cid:105)(cid:48)
x the long-run mean. We denote the macroeconomic state variables as, Ymac = θ p n ,
t t t t
and the set of unknown parameters are θ,ρ ,m ,m ,n,σ ,σ ,ρ , σ ,ρ , and σ .
θ p n θp θn p pp n nn
In this model, the conditional mean has an autoregressive component, but macro risks
can also affect expected growth. This can both accommodate cyclical effects (lower conditional
means in bad times), or the uncertainty effect described in Bloom (2009). The shocks reflect
the BEGE framework of Bekaert and Engstrom (2017), implying that the conditional higher
moments of output growth are linear functions of the bad and good uncertainties. For example,
the conditional variance and the conditional unscaled skewness are as follows,
(cid:104) (cid:105)
Conditional Variance: E (cid:0)uθ (cid:1)2 = σ2 p +σ2 n ,
t t+1 θp t θn t
(cid:104) (cid:105)
Conditional Unscaled Skewness: E (cid:0)uθ (cid:1)3 = 2σ3 p −2σ3 n .
t t+1 θp t θn t
This reveals the sense in which p represents “good” and n “bad” volatility: p (n ) increases
t t t t
(decreases) the skewness of industrial production growth.
Theindustrialproductionprocessisakeydeterminantoftheconsumptiongrowthprocess,
but we model consumption growth jointly with the cash flow processes for equities imposing
the economic restriction that those processes are cointegrated.
2.2.2 Cash Flows and Cash Flow Uncertainty
To model the cash flows for equites and corporate bonds, we focus attention on two vari-
ables that exhibit strong cyclical movements, namely earnings (see e.g. Longstaff and Piazzesi,
2004) and corporate defaults (see e.g. Gilchrist and Zakrajˇsek, 2012).
Corporate Bond Loss Rate To model corporate bonds, we use data on default rates.
Suppose a portfolio of one-period nominal bonds has a promised payoff of exp(c) at (t+1), but
will in fact only pay an unknown fraction F ≤ 1 of that amount. Let l = ln(1/F ) ≥ 0 be
t+1 t t
the log loss function. Then the actual nominal payment will be exp(c−l ). We use default
t+1
data on corporate bonds to measure this loss rate and provide more detail on the pricing of
defaultable bonds in the pricing section (Section 2.3).
The log loss rate, l , is defined as the logarithm of the current aggregate default rate
t
multiplied by the loss-given-default rate. The dynamic system of the corporate bond loss rate
is modeled as follows:
l = l +ρ l +ρ p +ρ n +σ ω +σ ω +ul (14)
t+1 0 ll t lp t ln t lp p,t+1 ln n,t+1 t+1
ul = σ ω (15)
t+1 ll l,t+1
6
ωl,t+1 ∼ Γ(cid:101)(vt,1), (16)
where
v = v +ρ v +σ ω . (17)
t+1 0 vv t vl l,t+1
The conditional mean depends on an autoregressive term and the good and bad uncertainty
state variables p and n . The loss rate total disturbance is governed by three indepen-
t t
dent heteroskedastic centered gamma shocks: the good and bad environment macro shocks
{ω ,ω } and the (orthogonal) loss rate shock ω . The loss rate shock follows a cen-
p,t+1 n,t+1 l,t+1
tered gamma distribution where the shape parameter v varies through time.
t
This dynamic system allows macro-economic uncertainty to affect both the conditional
mean and conditional variance of the loss rate process. However, it also allows the loss rate
to have an autonomous autoregressive component in its conditional mean (making l a state
t
variable) and accommodates heteroskedasticity not spanned by macro-economic uncertainty.
Therefore, v can be viewed as “financial” cash flow uncertainty. Note that the shock to v
t t
is the same as the shock for the loss process itself. If σ and σ are positive, as we would
ll vl
expect, the loss rate and its volatility are positively correlated; that is, in bad times with a high
incidenceofdefaults, thereisalsomoreuncertaintyaboutthelossrate, andbecausethegamma
distribution is positively skewed, the (unscaled) skewness of the process increases. We would
also expect the sensitivities to the good (bad) environment shocks, σ (σ ) to be negative
lp ln
(positive): defaults should decrease (increase) in relatively good (bad) times.
Theconditionalvarianceofthelossrateisσ2p +σ2 n +σ2v ,anditsconditionalunscaled
lp t ln t ll t
(cid:16) (cid:17)
skewness is 2 σ3p +σ3 n +σ3v . The set of unknown parameters are l , ρ , ρ , ρ , σ ,
lp t ln t ll t 0 ll lp ln lp
σ , σ , v , ρ , and σ .
ln ll 0 vv vl
Log Earnings Growth Log earnings growth, g , is defined as the change in log real earnings
t
of the aggregate stock market. It is modeled as follows:
g = g +ρ g +ρ(cid:48) Ymac+σ ω +σ ω +σ ω +ug (18)
t+1 0 gg t gy t gp p,t+1 gn n,t+1 gl l,t+1 t+1
ug = σ ω (19)
t+1 gg g,t+1
ω ∼ N(0,1). (20)
g,t+1
The conditional mean is governed by an autoregressive component and the three macro fac-
tors; the time variation in the conditional variance comes from the good and bad uncertainty
factors, and the loss rate uncertainty factor. The earnings shock is assumed to be Gaussian
and homoskedastic, which cannot be rejected by the data in our sample.1 A key implicit as-
sumption is that the conditional variance of earnings growth is spanned by macro-economic
1More specifically, we conduct the Kolmogorov-Smirnov test for Gaussianity and the Engle test for het-
eroscedasticity using the residuals of log earnings growth ug (this section), log consumption-earnings ratio uκ
(later), and log dividend-earnings ratio uη (later). We fail to reject the null that the residuals series, after
controlling for heteroskedastic fundamental shocks, are Gaussian and homoskedastic.
7
uncertainty and the financial uncertainty present in default rates. The set of unknown param-
eters is {g ,ρ ,ρ(cid:48) ,σ ,σ ,σ ,σ }.
0 gg gy gp gn gl gg
Log Consumption-Earnings Ratio We model consumption as stochastically cointegrated
with earnings so that the consumption-earnings ratio becomes a relevant state variable. Define
(cid:16) (cid:17)
κ ≡ ln Ct which is assumed to follow:
t Et
κ = κ +ρ κ +ρ(cid:48) Ymac+σ ω +σ ω +σ ω +uκ (21)
t+1 0 κκ t κy t κp p,t+1 κn n,t+1 κl l,t+1 t+1
uκ = σ ω (22)
t+1 κκ κ,t+1
ω ∼ N(0,1). (23)
κ,t+1
Similarly to earnings growth, there is an autonomous conditional mean component but the
heteroskedasticity of κ is spanned by other state variables. The set of unknown parameters is
t
{κ ,ρ ,ρ(cid:48) ,σ ,σ ,σ ,σ }.
0 κκ κy κp κn κl κκ
Log Dividend Payout Ratio The log dividend payout ratio, η , is expressed as the log ratio
t
of dividends to earnings. Recent evidence in Kostakis, Magdalinos, and Stamatogiannis (2015)
shows that the monthly dividend payout ratio is stationary. We model η analogously to κ and
t t
g :
t
η = η +ρ η +ρ(cid:48) Ymac+σ ω +σ ω +σ ω +uη (24)
t+1 0 ηη t ηy t ηp p,t+1 ηn n,t+1 ηl l,t+1 t+1
uη = σ ω (25)
t+1 ηη η,t+1
ω ∼ N(0,1). (26)
η,t+1
The set of unknown parameters is {η ,ρ ,ρ(cid:48) ,σ ,σ ,σ ,σ }.
0 ηη ηy ηp ηn ηl ηη
2.2.3 Pricing Kernel State Variables
Inthemodelweintroducedabove, therealpricingkerneldependsonconsumptiongrowth
and changes in risk aversion. To price nominal cash flows (or to price default free nominal
bonds), we also need an inflation process. We discuss the modeling of these variables here.
(cid:16) (cid:17)
Consumption Growth By definition, log real consumption growth, ∆c = ln Ct+1 =
t+1 Ct
g +∆κ . Therefore, consumption growth is spanned by the previously defined state vari-
t+1 t+1
ables and shocks.
(cid:16) (cid:17)
Risk Aversion The state variable capturing risk aversion, q ≡ ln Ct is, by definition,
t Ct−Ht
nonnegative. We impose the following structure,
q = q +ρ q +ρ p +ρ n +σ ω +σ ω +uq (27)
t+1 0 qq t qp t qn t qp p,t+1 qn n,t+1 t+1
uq = σ ω (28)
t+1 qq q,t+1
8
ωq,t+1 ∼ Γ(cid:101)(qt,1). (29)
Theriskaversiondisturbanceiscomprisedofthreeparts,exposuretothegooduncertainty
shock, exposure to the bad uncertainty shock, and an orthogonal preference shock. Thus,
given the distributional assumptions on these shocks, the model-implied conditional variance is
σ2 p +σ2 n +σ2 q ,andtheconditionalunscaledskewness2(cid:0)σ3 p +σ3 n +σ3 q (cid:1). Wemodel
qp t qn t qq t qp t qn t qq t
the pure preference shock also with a demeaned gamma distributed shock, so that its variance
and (unscaled) skewness are proportional to its own level. Controlling for current business
conditions, when risk aversion is high, so is its conditional variability and unscaled skewness.
The higher moments of risk aversion are perfectly spanned by macroeconomic uncertainty on
the one hand and pure sentiment (q ) on the other hand. Note that our identifying assumption
t
is that q itself does not affect the macro variables and u represents a pure preference
t q,t+1
shock. The conditional mean is modeled as before: an autonomous autoregressive component
and dependence on p and n . The set of unknown parameters describing the risk aversion
t t
process is {q ,ρ ,ρ ,ρ ,σ ,σ ,σ }.
0 qq qp qn qp qn qq
Inflation Topricenominalcashflowsandnominalbonds,wemustspecifyaninflationprocess.
Theconditionalmeanofinflationdependsonanautoregressivetermandthethreemacrofactors
Ymac. The conditional variance and higher moments of inflation are proportional to the good
t
and bad uncertainty factors {p ,n }. The inflation innovation uπ is assumed to be Gaussian
t t t+1
and homoskedastic. There is no feedback from inflation to the macro variables:
π = π +ρ π +ρ(cid:48) Ymac+σ ω +σ ω +uπ (30)
t+1 0 ππ t πy t πp p,t+1 πn n,t+1 t+1
uπ = σ ω (31)
t+1 ππ π,t+1
ω ∼ N(0,1). (32)
π,t+1
The set of unknown parameters is {π ,ρ ,ρ(cid:48) ,σ ,σ ,σ }.
0 ππ πy πp πn ππ
2.2.4 Matrix Representation
The dynamics of all state variables introduced above can be written compactly in matrix
(cid:104) (cid:105)(cid:48)
notation. WedefinethemacrofactorsYmac = θ p n andotherstatevariablesYother =
t t t t t
(cid:104) (cid:105)(cid:48)
π l g κ η v q . Among the ten state variables, the industrial production growth
t t t t t t t
θ , the inflation rate π , the loss rate l , earnings growth g , the log consumption-earnings ratio
t t t t
κ and the log divided payout ratio η are observable, while the other four state variables,
t t
{p ,n ,v ,q } are latent. There are eight independent centered gamma and Gaussian shocks in
t t t t
this economy. The system can be formally described as follows (technical details are relegated
to the Appendix):
Y = µ+AY +Σω , (33)
t+1 t t+1
where constant matrices, µ (10 × 1), A (10 × 10) and Σ (10 × 8), are implicitly defined,
(cid:104) (cid:105)(cid:48)
Y = Ymac(cid:48) Yother(cid:48) (10 × 1) is a vector comprised of the state variable levels, and
t t t
9